What is Gamma function: Definition and 128 Discussions

In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For any positive integer n,




{\displaystyle \Gamma (n)=(n-1)!\ .}
Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral:











{\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx,\ \qquad \Re (z)>0\ .}
The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.
The gamma function has no zeroes, so the reciprocal gamma function




{\displaystyle 1/\Gamma }
is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function:







{\displaystyle \Gamma (z)={\mathcal {M}}\{e^{-x}\}(z).}
Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.

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    I Relation Between Beta and Gamma functions

    So, my teacher showed me this proof and unfortunately it is vacation now. I don't understand what just happened in the marked line. Can someone please explain?
  2. nomadreid

    I Applications of complex gamma (or beta) functions in physics?

    An example of physical applications for the gamma (or beta) function(s) is http://sces.phys.utk.edu/~moreo/mm08/Riddi.pdf (I refer to the beta function related to the gamma function, not the other functions with this name) The applications in Wikipedia...
  3. T

    I Closed Form for Complex Gamma Function

    Hey all, I was wondering if there was an equivalent closed form expression for ##\Gamma(\frac{1}{2}+ib)## where ##b## is a real number. I came across the following answer...
  4. PhysicsRock

    Exploring an Unexplored Expression for the Beta Function

    So, I've recently played around a little with the Gamma Function and eventually managed to find an expression for the Beta Function I have not yet seen. So I'm asking you guys, if you've ever seen this expression somewhere or if this is a new thing. Would be cool if it was, so here's the...
  5. benorin

    Relief of Complex Gamma Fcn — was this hand drawn?

    This pic is from an older text called Tables of Higher Functions (interestingly both in German first then English second) that I jumped at buying from some niche bookstore for $40. Was this hand drawn? I think I’ve seen was it that mathegraphix or something like that linked by @fresh_42...
  6. benorin

    I Functions.wolfram.com accepted and published my formula for Gamma Fn

    I sent them an email about a week or so ago with the images of the following from the solutions to exercises section 1 of my Insight Article A Path to Fractional Integral Representations of Some Special Functions: 1.9) Use partial fraction decomposition and the Euler limit definition of the...
  7. patric44

    I Integral representation of incomplete gamma function

    hi guys I was trying to verify the integral representation of incomplete gamma function in terms of Bessel function, which is represented by $$\gamma(a,x) = x^{\frac{a}{2}}\;\int_{0}^{∞}e^{-t}t^{\frac{a}{2}-1}J_{a}(2\sqrt{xt})dt\;\;a>0$$ i was thinking about taking substitutions in order to...
  8. M

    Showing that this identity involving the Gamma function is true

    My attempt at this: From the general result $$\int \frac{d^Dl}{(2\pi)^D} \frac{1}{(l^2+m^2)^n} = \frac{im^{D-2n}}{(4\pi)^{D/2}} \frac{\Gamma(n-D/2)}{\Gamma(n)},$$ we get by setting ##D=4##, ##n=1##, ##m^2=-\sigma^2## $$-\frac{\lambda^4}{M^4}U_S \int\frac{d^4k}{(2\pi)^4} \frac{1}{k^2-\sigma^2} =...
  9. Pouramat

    QED, chapter 4 P&S page 125 (Coulomb Potential)

    $$\bar u(p') \gamma^i u(p) = u^\dagger(p') \gamma^0 \gamma^i u(p)$$ if ##p = p'## we can use $$u^\dagger(p) u(p) = 2m \xi^\dagger \xi$$ but how can we conclude the statement?
  10. O

    MHB Evaluate some kind of gamma function

    My question and solution that I've tried out are in attachment. Is it true my steps?
  11. W

    Gamma Function Q from Mary Boas 2nd ed (ch11)

    So using $$L=\frac{mv^2}{2} - \frac{1}{2} m lnx$$ and throwing it into the Euler-L equation I agree with kcrick & OlderDan that we can manipulate this to either $$\frac{d}{dt} m\dot{x} = -\frac{m}{2x}$$ or $$2vdv = -\frac{dx}{x}$$ but I'm not having any epiphanies on how to turn the above into...
  12. hilbert2

    A Analytical function from numerable point set

    Sometimes there are functions that are initially defined for only integer values of the argument, but can be extended to functions of real variable by some obvious way. An example of this is the factorial ##n!## which is extended to a gamma function by a convenient integral definition. So, if I...
  13. cg78ithaca

    A Generalization of hypergeometric type differential equation

    I am aware that hypergeometric type differential equations of the type: can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, and λ is a constant. I'm trying to reproduce the method for the case where λ is not constant...
  14. M

    MHB Gamma function is convergent and continuous

    Hey! :o I want to show that the Gamma function converges and is continuous for $x>0$. I have done the following: The Gamma function is the integral \begin{equation*}\Gamma (x)=\int_0^{\infty}t^{x-1}e^{-t}\, dt\end{equation*} Let $x>0$. It holds that...
  15. L

    A Gamma function, Bessel function

    I have question regarding gamma function. It is concerning ##\Gamma## function of negative integer arguments. Is it ##\Gamma(-1)=\infty## or ##\displaystyle \lim_{x \to -1}\Gamma(x)=\infty##? So is it ##\Gamma(-1)## defined or it is ##\infty##? This question is mainly because of definition of...
  16. L

    A Gamma function convergence of an integral

    ##\Gamma(x)=\int^{\infty}_0 t^{x-1}e^{-t}dt## converge for ##x>0##. But it also converge for negative noninteger values. However many authors do not discuss that. Could you explain how do examine convergence for negative values of ##x##.
  17. binbagsss

    How Do You Solve Gamma Function Identities?

    Homework Statement To show: Homework Equations The Attempt at a Solution To be honest, I'm pretty stuck. I could try to use the third identity: ##\Gamma(-k+\frac{1}{2})=\frac{2\sqrt{\pi}}{2^{-2k}}\frac{\Gamma(-2k)}{\Gamma(-k)} ## but this doesn't really seem to get me anywhere. I...
  18. Mr Davis 97

    Integral using the gamma function

    Homework Statement I am trying to evaluate the following integral: ##\displaystyle \int^{\infty}_0 (1 - e^{-x}) x^{-\frac{3}{2}} \, dx## Homework EquationsThe Attempt at a Solution When I split the above integral, I get the following ##\int^{\infty}_0 x^{-\frac{3}{2}} \, dx - \Gamma...
  19. binbagsss

    Show two inequalities - (context gamma function converges)

    Homework Statement I'm not after another proof. I've just got a couple of inequalities I don't know how to show when following a given proof in my book. These are: Q1) ## 0\leq x \leq 1 \implies x^{t-1} e^{-x} \leq x^{t-1} ## So this is obvioulsy true, however I think I'm being dumb because...
  20. dykuma

    Sterling approximation of Beta Function

    Homework Statement Homework Equations The Attempt at a Solution I think this problem is probably a lot simpler than I am making it out to be. However, when I apply sterling's approx., I get a very nasty equation that does not simplify easily. One of the biggest problems I have though is...
  21. R

    I Gamma Function Explained for Upperclassmen Math Students

    I took differential equation over the last year and we talked about gamma functions in class but it wasn't in our books and I don't speak broken Russian so it was hard to understand what was going on. I'm wondering if the gamma function is similar to the dirac or heaviside functions and if...
  22. M

    MHB Using gamma function to solve an integral

    Hello, I have attached a picture of the integral I am solving. I understand at the end how they turned the function (in the second to last step) into gamma(2) BUT what I do not understand is how you can simply just remove the (2/theta) out of the exponent of e, turn it into gamma(2) then divide...
  23. S

    Volume in n Dimensions: Understanding the Meaning of n=0

    Hello, Surfing across the internet, I learned that the volume of a sphere in n dimensions can be expressed by V(n) = (Π^(n/2)) / Γ((n/2)+1), where n is the number of dimensions we are considering But if we consider n=0, then we get 1. So, how do we interpret this? I mean what does volume in zero...
  24. Muthumanimaran

    How Does the Gamma Function Simplify to 'z' in the Limit?

    This is given in Mathematical methods for physicists by Arfken and weber, while defining a property of gamma function, I have no idea how the term in the red circle becomes 'z' in the final step, please help
  25. davidbenari

    Writing complicated integral in terms of the Gamma function

    Homework Statement Write ##\int_{0}^{1}x^2(ln\frac{1}{x})^3 dx## in terms of the gamma function 2. Relevant equation ##\Gamma(p+1)=p\Gamma(p)## The Attempt at a Solution Say ##x=e^{-u}## one would eventually obtain the integral ##\int_{-\infty}^{0} u^3 e^{-u} du## STEPS: ##x=e^{-u}##...
  26. David Carroll

    How to evaluate the gamma function for non-integers

    Hello, everyone. After my discovery some time ago of the gamma function \int_a^b x^{-n}e^{-x}dx (where b = infinity and a = 0...sorry, haven't quite figured out LaTex yet...and actually the foregoing is the factorial function [I think it's silly that the argument has to be shifted down by...
  27. E

    Write this integral in terms of gamma function

    Hey! So I'm self studying mary boas's mathematical methods book and I've come across this integral: \int _{0}^{\infty }e^{-x^4}dx and I'm suppose to write this using the gamma function. The hint given states to let x^4 = u. And the answer is: \Gamma \left( \dfrac {5} {4}\right) I tried...
  28. C

    Contour integral involving gamma function

    Homework Statement Evaluate the integral by closing a contour in the complex plane $$\int_{-\infty}^{\infty} dx e^{iax^2/2}$$ Homework Equations Residue theoremThe Attempt at a Solution My initial choice of contour was a semicircle of radius R and a line segment from -R to R. In the limit R to...
  29. H

    The factorial of a rational number, the gamma function not used

    My first question is: is this formula (at the bottom) a known formula? In this subject i haven't explained how i build up the formula. So far i think it is equal to the gamma function of Euler with \Gamma\left(\frac{m_1}{m_2}+1\right)= \frac{m_1}{m_2}\ ! with m_1 , m_2 \in...
  30. mesa

    Looking for Borwein's/Zucker's fast algorithm for the gamma function.

    I have heard that the Borwein/Zucker algorithm for computing certain values of the gamma function is pretty awesome, but finding it online is proving elusive... Does anyone know the algorithm?
  31. Greg Bernhardt

    Can the Gamma Function Simplify Complex Integrals?

    Definition/Summary The gamma function denoted by \Gamma (n) is defined by \Gamma (n) = \int_{0}^{\infty} x^{n-1} e^{-x} dx is convergent for real and complex argument except for 0, -1, -2, ...-k Equations Useful identities: \Gamma(n+1)=n! \Gamma (x+1) = x\Gamma(x)...
  32. P

    How to simplify this summation to an incomplete gamma function

    Could someone please explain why the following sum simplifies to the following? = As far as I can see, this sum does not correlate to the formula for incomplete gamma function as a sum. I'd appreciate any help as the incomplete gamma function is somewhat beyond the scope of my current...
  33. Saitama

    MHB Solving a definite integral without using gamma function

    Problem: Evaluate: $$\int_0^{\infty} t^{-1/4}e^{-t}\,dt$$ Attempt: I recognised this one as $\Gamma(3/4)$. I found a few formulas on Wolfram Mathworld website which helps to evaluate this but I am wondering if I can solve the definite integral from elementary methods (like by parts). Any help...
  34. E

    Gamma Function and the Euler-Mascheroni Constant

    I was taking a break from studying from my real analysis, electrodynamics, and nuclear physics exams this week, and, being a math-phile, I decided to play around with the gamma-function for some reason. Anyway, I used the common product expansion of the multiplicative inverse, and I arrived at a...
  35. mesa

    Need sources to search for gamma function infinite series identities.

    I have a hard time believing we only have the limited number of series I have seen so far especially considering how much broader mathematics is than I had thought just a short while ago. Where should I search to find more infinite series summations for the gamma function? For example which...
  36. mesa

    Do we have a general solution infinite series for the gamma function?

    Does anyone know if we currently have an infinite series summation general solution for the gamma function such as, $$\frac{1}{\Gamma(k)}=\sum_{n=0}^{\infty} f(n,k)$$ or, $${\Gamma(k)}=\sum_{n=0}^{\infty} f(n,k)$$?
  37. H

    Representing the Gamma Function

    I'm not sure if this is a "general" math question but I do think it is an interesting one. The Gamma Function, \Gamma(t), has many interesting definitions. It can take on the form of an integral to an infinite product. There is one particular definition, however, that I am trying to...
  38. A

    Puzzle with moment generating function for gamma function

    I am given that The kth moment of an exponential random variable with mean mu is E[Y^k] = k!*mu^k for nonnegative integer k. I found m^2 (0) = (-a)(-a-1)(-beta)^2. The answer I found is however mu^2+a*beta^2 which is different from the k! From the given formula. Could someone help me figure it...
  39. R

    Gamma Func Help: Solve F(alpha+1)=alpha*F(alpha)

    http://img202.imageshack.us/img202/3224/620u.jpg In A for this question F(alpha+1)=alpha*F(alpha) and I'm curious as to how the RHS of this still has the gamma function in it?
  40. C

    Integrating a x^k ln(x) Function with Gamma Function

    Homework Statement "Show that - \int^1_0 x^k\ln{x}\,dx = \frac{1}{(k+1)^2} ; k > -1. Hint: rewrite as a gamma function. Homework Equations Well, I know that \Gamma \left( x \right) = \int\limits_0^\infty {t^{x - 1} e^{ - t} dt}. The Attempt at a Solution I've tried various substitutions...
  41. T

    Approximating the gamma function near x=-3

    I've just started self studying James Nearing's "Mathematical Tools for Physicists" (available at http://www.physics.miami.edu/~nearing/mathmethods/mathematical_methods-three.pdf), and I'm having trouble with problem 1.16 about the gamma function, defined for positive x as \Gamma(x)=...
  42. B

    Factorial Notation for Gamma Function Expression with Fractional Input?

    Hi members, gamma (n+1)=n! Now I have the following: gamma(2(n+3/2))=gamma(2n+1) What is factorial notation for this?? Thank you
  43. Mandelbroth

    Trouble with Calculating Gamma Function Integral Convergence on Wolfram Alpha?

    This is mostly calculus, but the question is computer based, I think. The antiderivative of the gamma function is, fairly trivially, ##\displaystyle \int_{0}^{+\infty}\frac{t^{z-1}}{e^{t}\ln{t}}-C##, where C is an arbitrary constant. Why does Wolfram Alpha have trouble calculating the...
  44. D

    Question regarding gamma function

    Homework Statement I have a quick question about the gamma function. According to my textbook it says: \int_{0}^{\infty }{{{x}^{n}}{{e}^{-ax}}dx}=\frac{\Gamma \left( n+1 \right)}{{{a}^{n+1}}} where \Gamma is the gamma function. My question is, do n has to be an integer number, or can it also...
  45. G

    Particular Values of Gamma Function

    Hi, I would like to know why is it that gamma(1/2) = squareroot(pi) The other values as well. I don't see why out of all values that the gamma function converges to operations of pi of all things. Is there some kind of relationship to the gamma function and the trigonometric function...
  46. G

    How does WolframAlpha use the gamma function to solve this integral?

    Hi I have this integral that I want to express in terms of a gamma function. Unfortunately I am unable to bring it in this form. So can you give me a hint how wolframalpha does thishttp://m.wolframalpha.com/input/?i=∫e%5E%28ix%29%2F%28ix%29%5E%281%2F5%29dx+from+-+infinity+to+infinity&x=10&y=3
  47. E

    Solving the Gamma Function: Using Recursion & Tables

    Homework Statement Questions are in picture. Homework Equations $$ \int _{0}^{\infty }x^{n}e^{-x}dx $$ = $$ Gamma (n+1) = n! $$ Gamma(P+1) $$ = $$Gamma(P)$$ $$ Gamma(P) = (1/P) $$Gamma(P+1)$$ The Attempt at a Solution 2) I have found it from table. 3) I have used recursion and...
  48. N

    Behaviour of Gamma function when z = -n

    Hi there, I'm actually trying to understand why the behaviour of the Gamma function at z = -n is (-1)^n/(n!z) + O(1) The first term (although without the z) I recognized it as the residue of f when z= -n. But the rest, no idea. Any explanation is very appreciated.
  49. Z

    How do you derive this alternate form of the gamma function?

    \Gamma(n) = int(0 to infinity)[(x^(n-1))*e^-x]dx Show that it can also be written as: \Gamma(n) = 2int(0 to infinity)[(x^(2n-1))*e^(-x^2)]dxI have no idea how to go about this. I have tried integration by parts of each to see if anything relates, but how can you get from an exp(-x) to and...